In order to determine the other cosmological parameters from the
supernova data we must consider supernova at large distances (z 0.3). Just as large
distance measurements on Earth show us the
curvature (geometry) of Earth's surface, so do large distance
measurements in cosmology show us the geometry of the universe. Since,
as we have seen, the geometry of the universe depends on the values of
the cosmological parameters, measurements of the luminosity distance for
distant supernova can be used to extract these values.

To obtain the general expression for the luminosity distance, consider
photons from a distance source moving radially toward us. Since we are
considering photons, ds2 = 0, and since they are
moving radially,
d2 =
d2 =
0. The Robertson-Walker metric, Eq. (1), then reduces to
0 = c2dt2 - a2dr2(1 - k r2)-1, which
implies

(31)

To get another expression for dt, we multiply Eq. (17) by
a(t)2 which produces an expression for
(da / dt)2. Furthermore, we note
that since the universe is expanding, the matter density is a function
of time. Given that lengths scale as a(t), volumes scale
as a(t)3 and therefore,

(32)

Using these facts, together with the definitions of the density
parameters in Eq. (22), Eq. (17) becomes

(33)

As previously mentioned, it is better to write things in terms of
measurable quantities, and in this case we can directly relate the
cosmic scale factor to the redshift z. The redshift is defined such
that

(34)

where 0 is
the current (received) value of the wavelength and
is
the wavelength at the time of emission. The redshift is a direct result
of the cosmic expansion and it can be shown that
[14]
a(t) ;
therefore,

(35)

Using Eq. (35) and the fact that
k = 1 -
m -
from Eq. (21), Eq. (33) can be rewritten as

(36)

Equating the expressions in Eqs. (31) and (36) and integrating, leads to
an expression for the radial coordinate r of the star. The luminosity
distance is then given by
[15]
d = (1 + z) a0r . Therefore,

(37)

where sinn(x) is sinh(x) for k < 0,
sin(x) for k > 0, and if k = 0 neither sinn nor
|k, 0|
appear in the expression. We
see that the functional dependence of the luminosity distance is
d (z;
m,
).

Inserting Eq. (37) into Eq. (24), and using the intercept from Eq. (28),
we get a redshift-magnitude relation valid at high z

(38)

In practice, astronomers observe the apparent magnitude and redshift of
a distant supernova. The density parameters are then determined by those
values that produce the best fit to the observed data according to
Eq. (38) for different cosmological models.

Under the continued assumption that the fluid pressure of the matter in
the universe is negligible (pm
0), Eq. (16) implies
that the deceleration parameter at the present time is given by

(39)

Therefore, once the density parameters have been determined by the above
procedure, the deceleration parameter can then be found.

Figure 2 illustrates how high-redshift data can
be used to estimate the
cosmological parameters and provide evidence in favor of a nonzero
cosmological constant. In this figure, the abscissa is the difference
between the distance moduli for the observed supernovae and what would
be expected for a traditional cosmological model such as those
represented in Table 1. The case shown
is based on the data of Riess
et. al. [16]
using a traditional model with
m = 0.2 and
= 0
represented by the central line
(m -
M) = 0. The figure shows that the data points lie predominantly
above the
zero line. This result means that the supernovae are further away (or
equivalently, dimmer) than traditional, decelerating cosmological models
allow. The conclusion then is that the universe must be accelerating. As
suggested by Eq. (39), the most straightforward explanation of this
conclusion is the presence of a nonzero, positive cosmological
constant. The solid curve, above the zero line in
Fig. 2, represents a
best-fit curve to the data that corresponds to a universe with
m = 0.24 and
=
0.72.

Figure 2. Using high-redshift data to
determine cosmological parameters
and provide evidence for a nonzero cosmological constant. The zero line
corresponds to a traditional decelerating model of the universe with
m = 0.2,
=
0, and
k =
0.8. The data
points are the high-redshift supernovae from Ref. 16. The solid curve
corresponds to those cosmological parameters that produce a best fit to
the data points as determined in Ref. 16.

Typical values for the cosmological parameters as determined by detailed
analysis of the type just discussed are the following:
[16]

(40)

Note that the negative deceleration parameter is consistent with an
accelerating universe. Furthermore, these values imply that the universe
is effectively flat predicting a curvature parameter roughly centered
around
k 0.04.